Inverted tooth chains 10 have long been used to transmit power and motion between shafts in automotive applications and as shown in FIG. 1, they are conventionally constructed as endless chains with ranks or rows 30a,30b, etc. of interleaved link plates 30 each with a pair of teeth 34 having outside flanks 37, and inside flanks 36 between the teeth defining a crotch 35, and each having two apertures 32 that are aligned across a link row to receive connecting pins 40 (e.g., round pins, rocker joints, etc.) to join the rows pivotally and to provide articulation of the chain 10 about pin centers C as it drivingly engages the sprocket teeth either at the inside flanks (“inside flank engagement”) or at the outside flanks (“outside flank engagement”) of the link plates at the onset of meshing with the driving and driven sprockets. The pin centers C are spaced at a chain link pitch P. The term “pin centers C” is intended to encompass the axis of rotation of successive link rows 30a,30b relative to each other, regardless of whether the pins 40 comprise round pins, rocker joints or another suitable joint. The outside flanks 37 are straight-sided (but could be curved) and are defined by an outer or outside flank angle ψ. The inside flanks are convexly curved and comprise circular arc segments defined by a radius R centered at an arc center 79 (FIG. 3A).
Although both inside flank engagement and outside flank engagement meshing styles have been used for automotive engine timing drives, inside flank engagement is more common. Referring still to FIG. 1, inside flank meshing contact is facilitated by the outward projection λ of the leading (in terms of chain movement direction) inside flank 36 of a link plate 30 with respect to the outside flank 37 of an adjacent link plate 30 in a preceding row 30a when the link rows 30a,30b are positioned in a straight line as would nominally be the case in the unsupported chain span at the onset of meshing with a sprocket.
Chain-sprocket impact at the onset of meshing is a dominant noise source in chain drive systems and it occurs as a chain link row exits the span and impacts with a sprocket tooth at engagement. The complex dynamic behavior of the meshing phenomenon is well known in the art and the magnitude of the chain-sprocket meshing impact is influenced by various factors, of which polygonal effect (referred to as “chordal action” or “chordal motion”) is known to induce a transverse vibration in the “free” or unsupported span located upstream from the sprocket as the chain approaches the sprocket along a tangent line. Chordal motion occurs as the chain engages a sprocket tooth during meshing and it will cause chain motion in a direction perpendicular to the chain travel and in the same plane as the chain and sprockets. This undesirable oscillatory chain motion results in a velocity difference between the meshing chain link row and a sprocket tooth at the point of initial contact, thereby contributing to the severity of the chain-sprocket meshing impacts and the related chain engagement noise levels.
FIGS. 2A and 2B illustrate the chordal rise for a sprocket in which chordal rise CR is conventionally defined as the vertical displacement of a chain pin center C (or other chain joint) as it moves through an angle α/2, where:CR=rp−rc=rp[1−cos(180°/N)]and where rc is the chordal radius or the distance from the sprocket center to a sprocket pitch chord of length P, which is also equal to the chain pitch length; rp is the theoretical pitch radius of the sprocket, i.e., one-half of the pitch diameter PD; N is the number of sprocket teeth; and α is equal to the sprocket tooth angle or 360°/N. FIG. 2A shows the chain pin center C at a first position where it has just meshed with the sprocket and where it is simultaneously aligned with both the tangent line TL and the sprocket pitch diameter PD. As is known in the art, and as used herein, the tangent line TL is the theoretical straight-line path along which the meshing chain pin centers C approach the sprocket. As shown herein, the tangent line TL is located in a horizontal orientation, in which case the tangent line TL is tangent to the pitch diameter PD at the top-dead-center or 12 o'clock position on the pitch diameter PD, i.e., the tangent line TL is tangent to the pitch diameter PD at a location where a chain pin center is centered on the pitch diameter PD and is also centered on a radial reference line that is normal to the tangent line TL (the reference line being vertical when the tangent line is horizontal as shown herein). FIG. 2B illustrates the location of the same pin center C after the sprocket has rotated through the angle α/2, where it can be seen that the pin center C is transversely displaced by a distance CR as it continues its travel around the sprocket wrap, and this vertical displacement of the pin center results in a corresponding displacement of the upstream chain span and tangent line TL thereof. This transverse displacement of the chain pins C as they move through the chordal rise and fall serves to induce undesired vibration in the unsupported chain span.
One attempt to reduce undesired chordal motion of the chain is described in U.S. Pat. No. 6,533,691 to Horie et al. Horie et al. disclose an inverted tooth chain wherein the inside flanks of each link plate are defined with a compound radius profile intended to smooth the movement of the inside flanks from initial sprocket tooth meshing contact to the fully meshed (chordal) position. Initial meshing contact for the Horie et al link plate form occurs at a convexly arcuate portion of the inside flank at the link toe tip and proceeds smoothly and continuously to a second arcuate portion of the inside flank before transitioning to outside flank full meshing contact of a preceding link.
Chordal motion is also reduced in the system disclosed in U.S. Pat. No. 7,789,783 by Young et al, where the prominence of the inside flanks of the chain link plates relative to the respective outside flanks of link plates in adjacent link rows is defined as a function of the chain pitch P, and the maximum projection of the inside flank Lambda (λ) relative to the related outside flank is defined to fall in the range of 0.010×P≦λ≦0.020×P. Young et al disclose a link plate that also incorporates inside flank initial meshing contact to limit chordal motion, but its inside flank meshing contact begins and ends on the same convexly arcuate portion of the link plate before the meshing contact transitions to outside flank full meshing contact of a preceding link plate to complete the meshing cycle.
In U.S. Pat. No. 6,244,983, Matsuda discloses a link plate having inside flank meshing contact with the sprocket tooth for the full meshing cycle. Although the outside flanks of the Matsuda link plate do not contact the sprocket teeth, its inside flank meshing geometry serves to restrict chordal motion during engagement.
The above mentioned prior art inverted tooth chains all have features to beneficially limit chordal motion during meshing. However, another important factor to have an adverse influence on chain drive noise levels was not sufficiently considered in the link plate design for these chains—as well as for other prior art inverted tooth chains—and that factor is the meshing impact geometry during the chain-sprocket engagement process.
As shown in FIG. 3 and more clearly in FIG. 3A, a prior art chain link row 30c of chain 10 is at the onset of meshing with a sprocket tooth 60c of a conventional sprocket 50 in a chain drive system 15 including the chain 10, sprocket 50, and at least one other sprocket meshing with the chain 10. Reference will usually be made only to the individual chain link plates 30 visible in the foreground of each row 30a,30b,30c,etc., but those of ordinary skill in the art will recognize that the discussion applies to multiple link plates 30 across each row. Successive pin centers C are numbered C1, C2, C3, C4, etc. to distinguish them from each other.
The link row 30c is shown at the instant of initial meshing contact with a corresponding sprocket tooth 60c, i.e., at the instant of initial contact between the leading inside flank 36 of the chain link plate and the engaging flank 62c of the sprocket tooth 60c at an initial contact location IC on the engaging flank 62c. An initial contact angle Theta (θ) is defined between a first radial reference line L1 originating at the axis of rotation of the sprocket and extending normal to the tangent line TL and a second radial reference line TC originating at the axis of rotation of the sprocket and extending through the tooth center of the subject sprocket tooth 60c. At the instant of initial meshing impact IC for link row 30c, the preceding link row 30b exits the chain span and enters a “suspended state”, i.e., the link plates 30 of row 30b are not in direct contact with the sprocket 50 and are suspended between the meshing row 30c and a preceding row 30a that is in full meshing contact with a preceding sprocket tooth 60b. Link row 30b will remain in this suspended state as row 30c articulates through its sliding contact with the engaging flank 62c of sprocket tooth 60c from its initial meshing contact location IC to a final inside flank meshing contact location IF, at which time row 30b completes its meshing cycle and transitions into a position where its trailing outside flanks 37 make full meshing contact at location OF with sprocket tooth 60c (contact locations IF and OF are shown in FIGS. 4 and 4A). FIGS. 4 and 4A show the point in the meshing cycle referred to as “simultaneous meshing” in that link rows 30b and 30c are in simultaneous contact with sprocket tooth 60c, and with the next increment of sprocket rotation, link row 30c will separate from its inside flank meshing contact. Upon separation, link row 30c remains in the span, and it will enter the suspended state at the instant of initial meshing impact IC for a following row 30d with sprocket tooth 60d. 
It should be noted that prior to the instant of initial meshing impact for link row 30c (referring again to FIGS. 3 and 3A), the chain span effectively rotates about pin center C1 as row 30c articulates toward meshing impact IC with the sprocket tooth 60c. Thus, the pin center C1 can be referred to as the “controlling pin center.” The controlling pin center C1 is the closest preceding (downstream) pin center relative to the leading pin center C2 of the meshing link row 30c (the controlling pin center C1 is also the trailing pin center of the closest (in terms of chain travel direction) fully meshed link row 30a). As such, the following relationships are defined:                a meshing contact angle Tau (τ) is defined between the tangent line TL and an initial contact reference line 70 that passes through both the controlling pin center C1 and the initial contact location IC;        the initial contact reference line 70 defines a length L lever arm (FIG. 3A) between the controlling pin center C1 and the initial contact location IC;        a link plate entrance angle Beta (β) is defined between the initial contact reference line 70 and an inside flank reference line 74 that passes through the arc center 79 of the inside flank radius R and the initial contact location IC (the inside flank reference line 74 will be normal to the involute curve (or radial arc segment or other curved surface) of the engaging flank 62c of the sprocket tooth 60c);        a meshing impact angle Sigma (σ) is defined between the tangent line TL and the inside flank reference line 74, i.e., θ=τ+β.        
Chain-sprocket meshing impact results from a velocity difference between the meshing link row 30c and a sprocket tooth 60c at the initial contact location IC, and the related impact energy E generated as the sprocket tooth collects the meshing link row 30c from the chain span at the instant of initial meshing impact is defined by the equation:E=CmL2ω2 cos2(90−β)where C is a constant, m is equal to the mass of the single meshing link row 30c, L is the length from the controlling pin center C1 to the initial contact location IC, ω is the angular velocity of the sprocket, and β is the link plate meshing entrance angle. The meshing impacts along with the associated noise levels can be reduced by decreasing the velocity difference, which can be accomplished by reducing the meshing entrance angle β.
In addition, the impact energy E equation considers only the mass of the meshing link row 30c, and it does not take into account chain tension CT and this chain tension will add to the resultant meshing impact energy E and the associated overall noise levels. The chain tension CT will act on the sprocket tooth 60c at the onset of meshing and the tooth impact reaction force FS, equal and opposite to a link impact force FL, will vary with the magnitude of the meshing impact angle σ, where:
      F    S    =            F      H              cos      ⁢                          ⁢      σ      and where FH will be equal to CT in order to satisfy the summation of horizontal forces being equal to zero. These relationships are shown in FIGS. 3 and 3A (note that in FIG. 3A, the meshing impact angle Sigma (σ) and its component angles are shown relative to a reference line 72 that is parallel to the tangent line TL and extending through the initial contact location IC, coincident with the force vector FH). It should be noted that the sprocket tooth 60c, along with the next several teeth forward (downstream) of tooth 60c, share in the load distribution of the chain tension CT with the largest reaction force FH occurring at location IC of tooth 60c at the onset of initial meshing contact. The remaining portion of the chain tension loading acting on the several teeth forward of tooth 60c does not influence the meshing noise levels and is therefore not a consideration for this present development. To summarize, the link impact force vector FL acts at the meshing impact location IC during initial meshing contact and adds to the total meshing impact energy E and the related noise levels.
As described above, FIG. 4 shows simultaneous meshing contact, where the leading inside flanks 36 of link row 30c are contacting the engaging flank 62c of sprocket tooth 60c at location IF, and the trailing outside flanks 37 of preceding link row 30b are contacting the engaging flank 62c are location OF. FIG. 4A is a greatly enlarged partial view of FIG. 4 that also shows the forces resulting from the geometry of the simultaneous meshing contact phenomenon. This instant at which the tooth 60c transitions from “inside flank only” contact with leading inside flanks 36 of link row 30c to achieve simultaneous outside flank contact with trailing outside flanks 37 of preceding link row 30b can also be referred to as a transition point, and also defines the end of the meshing cycle for the tooth 60c, because the link row 30b is now fully meshed with both its leading and trailing pin centers C1,C2 located on the pitch diameter PD. A transition angle Phi (φ) is defined between the first radial reference line L1 and the second radial reference line TC marking the tooth center of tooth 60c. 
FIGS. 4 and 4A correspond respectively to FIGS. 3 and 3A, but relate to the transition phenomenon, and show that:                a transition contact angle Tau′ (τ′) is defined between the tangent line TL and a transition contact reference line 80 that passes through both the outside flank contact location OF and the controlling pin center C1 which, for the transition phenomenon, is the leading pin center of the link row transitioning to trailing outside flank contact at location OF (or the pin center C that is immediately preceding the pin center at the interface between the simultaneously meshing link rows);        the transition contact reference line 80 defines a length L′ lever arm between the controlling pin center C1 and the outside flank contact location OF;        a link plate transition angle Beta′ (β′) is defined between the transition contact reference line 80 and an outside flank reference line 84 that extends normal to the trailing outside flank 37 (the outside flank reference line 84 will also be normal to the involute curve (or radial arc segment or other curved surface) of the engaging flank 62c of the sprocket tooth 60c);        a transition impact angle Sigma′ (σ′) is defined between the tangent line TL and the outside flank reference line 84, i.e., σ′=τ′+β′.It should be noted that features in FIGS. 4 and 4A that correspond to features of FIGS. 3 and 3A are labeled with corresponding reference characters including a prime (′) designation, and not all are discussed further. Also, in FIG. 4A, the transition impact angle Sigma′ (σ′) and its constituents are shown relative to a reference line 82 that is parallel to the tangent line TL and extending through the outside flank contact location OF, coincident with the force vector F′H.        
The intensity of the secondary meshing impact and the related noise level as link row 30b transitions to full chordal meshing contact at location OF with sprocket tooth 60c is a smaller value as compared to the above-described initial meshing impact at location IC and its resulting meshing noise level. Firstly, the transition impact angle σ′ will always be a smaller value than the initial meshing impact angle σ. Secondly, the outside flank contact at location OF occurs as the link row 30b transitions from the suspended state to the fully meshed state, which is believed to be less significant in terms of impact force as compared to the initial contact between the chain 10 and sprocket 50, in which a link row is collected from the chain span to impact with a sprocket tooth 60 at the onset of meshing. In addition, noise and vibration testing has shown the transition meshing impact of the outside flank 37 at location OF to contribute less to the overall meshing noise levels than the initial meshing impact of the inside flank 36 at location IC.
The sprocket 50 is conventional and the teeth 60 (i.e., 60a, 60b, 60c, etc.) are each symmetrically defined about a radial tooth center TC to have an engaging flank 62 (i.e., 62a,62b,62c,etc.) that makes initial contact with the chain 10 during meshing and a matching disengaging flank 64 (i.e., 64a,64b,64c,etc.). The tooth centers TC bisect each tooth 60 and are evenly spaced in degrees)(°) at a tooth angle α=360°/N. The involute form of the engaging tooth flanks 62 (and disengaging flanks 64) is generated from a base circle and the base circle is defined as:                Base Circle=PD×COS(PA), where        PD=sprocket pitch diameter, and PA=tooth pressure angleFurthermore, the pitch diameter PD, itself, is defined as:        PD=P/SIN(180/N), where        where P=pitch, and N=number of teeth in sprocket.The involute tooth form can be approximated by a radial tooth form, and the pressure angle PA of a radial tooth form can likewise be determined. In any case, it is generally known that an engaging flank 62 defined with a smaller pressure angle is steeper (closer to a radial line originating at the sprocket axis of rotation) as compared to an engaging flank defined with a larger pressure angle. As such, a reference line tangent to the engaging flank 62 at the initial contact location IC will define an angle between itself and a radial reference line located between the engaging flank and the immediate downstream (leading) disengaging flank 64 that is smaller when the pressure angle is decreased and that is larger when the pressure angle is increased. Prior art systems have not substantially altered the conventional known sprocket tooth pressure angles to permit optimization of the design of the chain link plates 30 in order to minimize link impact force FL and the related impact energy E. Known sprocket pressure angles in degrees)(°) are shown below in TABLE 1, and the sprocket 50 conforms to these conventions (all teeth 60 have the same pressure angle PA):        
TABLE 1Sprocket ToothKnown SprocketCount (N)Pressure Angle<1934°19-2533°26-6031.5°